Stochastic Koopman Operators and DMD Algorithms Resistant to Noise

Oftentimes, complicated systems have a high degree of uncertainty; it can come from a lack of information on the system, noise in measurements, or true randomness in the evolution. Since a deterministic model fails to accurately represent the system, we can use the Stochastic Koopman Operator to predict the "expected value" of the future. When generating data-driven models of the system, standard Dynamic Mode Decomposition algorithms demonstrate a consistent bias. DMD can be shown to fail even for simple systems with a small amount of measurement noise in the data, With newer algorithms, we can generate models resistant to noise in data and randomness in the dynamics. We demonstrate a newer, more robust DMD algorithm which is resistant to noise in data. Additionally, it allows us to generate better models by introducing time shifts to the data similar to Hankel DMD, which fails when the system contains random elements.

Data-driven Generation of Quantum Accurate Interatomic Potentials for Hydrogen-Oxygen Combustion

Although quantum scale simulations of hydrogen-oxygen combustion offer an accurate description of the process, a multi-atom quantum simulation of combustion is unfeasible as it would not terminate in a scientist's lifetime. Multi-atom simulations of combustion are feasible at the molecular scale, however, the potential bond energies are inaccurate, and results often fail to match quantum data. We demonstrate how the programmable potentials methodology can be utilized to develop quantum accurate molecular level potentials for several intermediate reactions involved in hydrogen-oxygen combustion. Sparse Electronic Structure Theory (EST) simulation data is utilized to train our programmable potentials. The developed potentials are then inputted into the molecular dynamics simulation package LAMMPS for verification. Our results demonstrate that the developed programmable potentials generalize beyond the sparse EST training data set. Most importantly, the developed potentials lead to feasible and quantum-accurate molecular dynamics simulations of hydrogen-oxygen combustion.

Data-driven Analysis and forecasting of Highway Traffic Dynamics

The unpredictable elements involved in a vehicular traffic system, like human interaction and weather, lead to a very complicated, high-dimensional, nonlinear dynamical system. Therefore, it is difficult to develop a mathematical or artificial intelligence model that describes the time evolution of traffic systems. All the while, the ever-increasing demands on transportation systems has left traffic agencies in dire need of a robust method for analyzing and forecasting traffic. Here we demonstrate how the Koopman mode decomposition can offer a model-free, data-driven approach for analyzing and forecasting traffic dynamics. By obtaining a decomposition of data sets collected by the Federal Highway Administration and the California Department of Transportation, we are able to reconstruct observed data, distinguish any growing or decaying patterns, and obtain a hierarchy of previously identified and never before identified spatiotemporal patterns. Furthermore, it is demonstrated how this methodology can be utilized to produce real-time accurate forecasts for highways and networks of highways. 

Research Program Outline for Big Data Dynamics in Systems of Systems

The problem of design, reverse engineering and retrofitting for robust operation of large-scale interconnected dynamical systems is perhaps the engineering grand challenge of our time. Mathematics and engineering tools for treatment of individual components have been developed to a high degree of sophistication. However, when these components are connected - whether physically or by communication devices - new, collective phenomena can emerge that are not necessarily related to properties of individual components. The local consequences of such phenomena can be sensed - and the drive towards reduced cost and ubiquity of sensors leads to a massive amount of dynamically changing data. The phenomena indicated by sensed data have to be recognized, counteracted or perhaps even utilized dynamically in attempts to achieve optimal design and operation. Here are some of the critical elements of the applied problem at hand, the "Big Data Dynamics in Systems of Systems", and our viewpoint on the associated research directions.

New Operator Theoretical and Experimental Methods for Predicting Fundamental Mechanisms of Complex Chemical Processes

The complexity of chemistry stems from our inability to universally determine all the relevant dynamical processes involved in chemical transformations with an accurate accounting of electronic structure. Even relatively “simple” chemical reactions can be extremely complex when diverse reaction pathways connect many transient species. Robust methods are needed to predict how a set of reactants undergoes sequential, branching reactions, passing through many transition states and transient species, to reach a final set of stable products. 
Combustion Chemistry Project

Koopman operator theory and fluid mechanics

Nonlinear fluid flows represent some of the most complex nonlinear systems in the nature and the industry. This complexity is in part due to the nonlinearity of the governing equations and partly because of the high -and possibly infinite- number of dimensions required to model the fluid continuum. As a result, most of the quantitative prediction and analysis done today relies on extensive numerical simulations and experiments. Koopman operator theory offers a novel data-driven framework to extract dynamically-relevent information from the outputs of such numerical and experimental studies.

Applications of Koopman operator theory to hybrid systems

A considerable number of dynamical systems in engineering practice are essentially hybrid in nature. These systems model non-smooth phenomena such as impact, collision, and switching between discrete modes. The orbits of such systems are typically characterized by smooth evolutions, interrupted by discrete jumps, making the analysis of the state-space inherently more complicated. For these systems, the traditional interpretation of the state-space geometry in terms of individual trajectories may not offer much insight, and it is useful to consider the evolution of ensembles of initial conditions instead. In this research project, we take this “set-oriented” viewpoint to hybrid systems by characterizing their behavior through the spectral properties of the Koopman operator.